SPECTRAL CLUSTERING AND VISUALIZATION: A ... - Carl Meyer

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BENSON-PUTNINS, BONFARDIN, MAGNONI, <strong>AND</strong> MARTIN The next step in the Fiedler Method is to take each subgraph and partition it using its own Fiedler vector. For our example, this second iteration works well on the right hand part of the graph (see Figure 3.4). For this small graph, two iterations are sufficient to provide well connected clusters, but as graphs become larger, more iterations become necessary. There are different ideas for when one should stop partitioning the subgraphs. We created an algorithm that creates a specified number of clusters, k. Fig. 3.4. Partition made by the second iteration of the Fiedler Method 3.2. Limitations. The Fiedler Method has gained acceptance as a viable clustering technique for simple graphs. There are, however, some disadvantages to this method. First, the Fiedler Method is iterative, so if any questionable partitions are made, the mistake could be magnified through further iterations. Second, new eigendecompositions must be found at every iteration; this can be expensive for large data sets. Finally, this method was designed for undirected, weighted graphs. Unweighted graphs can be considered by assigning each edge to have weight 1. Directed graphs can be considered as well through utilization of additional eigenvectors of L, but we will not make use of these techniques. 4. MinMaxCut Method. Like the Fiedler Method, the MinMaxCut Method is a spectral clustering method that partitions a graph. Spectral clustering methods use the spectral, or eigen, properties of a matrix to identify clusters. There are a number of spectral clustering methods, several of which are given in detail in [12]. Although the Fiedler Method and the MinMaxCut Method are both spectral clustering methods, the MinMaxCut Method can create more than two clusters simultaneously while the Fiedler Method creates two clusters with each iteration. 4.1. Background. Before explaining the MinMaxCut Method, we describe a similar, more intuitive, algorithm: the Ratio Cut Method. The goal of this method is to partition an undirected, weighted graph into k clusters through the minimization of what is known as the ratio cut. Given a graph, such as a consensus matrix, broken into k clusters X 1 , X 2 , . . . , X k , the ratio cut is defined to be (4.1) k∑ i=1 w(X i , X i ) |X i | where |X| is the number of vertices in X, X is the complement of X and, given two subgraphs X and Y , w(X, Y ) is the sum of the weights of edges between X and Y . 6
CLUSTER <strong>AND</strong> DATA ANALYSIS Finding the minimum ratio cut of a graph by checking every possible collection of clusters is computationally prohibitive, but an approximation of the minimum can be found using linear algebra. Given a choice of k clusters in a graph with n vertices, let H be the n × k matrix with entries h ij = 1 if the ith vertex is in the jth √ |Xj| cluster, and 0 otherwise. Minimizing the ratio cut is then equivalent to minimizing T r(H T LH) over the set of matrices H whose columns form an orthonormal basis [12], where L is the Laplacian matrix defined in equation (3.1) and T r indicates the trace of a matrix. The conditions on H being constructed with entries based on the clusters can be relaxed to be H T H = I, i.e. the columns of H form an orthonormal set of vectors (see [13]). With this new condition, it is known that the minimum of T r(H T LH) over the set of matrices H whose columns form an orthonormal basis occurs when the columns of H are the eigenvectors of L corresponding to the k smallest eigenvalues [12]. With the original conditions for H, it is easy to determine which vertices are in the same cluster: the ith and jth vertices are clustered together if and only if the ith and jth rows of H are the same. The solution for H under the relaxed conditions does not have equal rows, but one can use a clustering technique to group the rows into k clusters. These clusters of rows correspond to clusters in the graph. Figure 4.1 shows the matrix H with both the original and relaxed conditions for the 10 vertex graph in Figure 3.1 with k = 3. With the original conditions, we can see that vertices 1, 8, and 9 cluster together, vertices 2, 3, and 7 cluster together, and vertices 4, 5, 6, and 10 cluster together. With the relaxed conditions, we cluster the rows of H using k-means with k = 3 and find the same clusters as before. This is consistent with the clustering found in Section 3.1, where the Fiedler Method was used to determine the clusters. Fig. 4.1. H with original conditions (left) and relaxed conditions (right) 4.2. Clustering with MinMaxCut Method. The algorithm we will use to cluster data, the MinMaxCut Method, is a variation of the Ratio Cut Method. We use the MinMaxCut Method because it creates clusters such that elements in the 7
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SPECTRAL CLUSTERING AND VISUALIZATION: A ... - Carl Meyer

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